Finite Group Databasebeta

← #31968097

Subgroups — #31968097

Order 18176.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598940C211yesmaximal, characteristic
2#22597491C211yesmaximal, characteristic
3#22587247C211yesmaximal, characteristic, complemented
4#22587240C211yesmaximal, characteristic, complemented
5#22587235C211yesmaximal, characteristic, complemented
6#22587230C211yesmaximal, characteristic, complemented
7#22587157C211yesmaximal, characteristic, complemented
8#22587150C211yesmaximal, characteristic, complemented
9C8×#11765855C211yesmaximal, characteristic, complemented
10C8×#11765852C211yesmaximal, characteristic, complemented
11#22587108C211yesmaximal, characteristic, complemented
12#22586798C211yesmaximal, characteristic
13#22586795C211yesmaximal, characteristic, complemented
14#22586621C211yesmaximal, characteristic, complemented
15C71×#2985C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17#17986405C411yescharacteristic
18#17986391C411yescharacteristic
19#17986367C411yescharacteristic
20#17986361C411yescharacteristic
21#17986255C411yescharacteristic
22#17986255C411yescharacteristic
23#17986247C411yescharacteristic
24#17986247C411yescharacteristic
25#17986129C2211yescharacteristic
26C8×Dic142C2211yescharacteristic
27C4×#11765855C2211yescharacteristic
28C4×#11765855C2211yescharacteristic
29C4×#11765852C2211yescharacteristic
30C4×#11765852C2211yescharacteristic
31#17986078C2211yescharacteristic
32#17986074—21no—
33#17986073—21no—
34C8×D284C2211yescharacteristic
35#17986059—21no—
36#17986024—21no—
37#17986023C2211yescharacteristic
38C4×#11765849C2211yescharacteristic
39C2×C4×C71⋊C8C2211yescharacteristic
40#17985751C2211yescharacteristic
41#17985751C2211yescharacteristic
42#17985749C2211yescharacteristic
43#17985749C2211yescharacteristic
44Q8⋊(C71⋊C8)C2211yescharacteristic
45Q8⋊(C71⋊C8)C2211yescharacteristic
46Q8⋊(C71⋊C8)C2211yescharacteristic
47Q8⋊(C71⋊C8)C2211yescharacteristic
48#17985729C2211yescharacteristic
49#17985729C2211yescharacteristic
50#17985726C2211yescharacteristic
51#17985726C2211yescharacteristic
52#17985721C2211yescharacteristic
53#17985721C2211yescharacteristic
54C8×C71⋊C8C2211yescharacteristic
55C8×C71⋊C8C2211yescharacteristic
56#533×C71C2211yescharacteristic
57#446×C71—21no—
58#446×C71—21no—
59#446×C71C2211yescharacteristic
60#446×C71C2211yescharacteristic
61C8×Q8×C71C2211yescharacteristic
62C8×Q8×C71C2211yescharacteristic
63#433×C71—21no—
64#433×C71—21no—
65#432×C71C2211yescharacteristic
66#431×C71C2211yescharacteristic
67C2×#11765855C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765852C2×C411yescharacteristic
70C2×#11765852C2×C411yescharacteristic
71C2×#11765849—21no—
72C2×#11765849C2×C411yescharacteristic
73C2×#11765849C2×C411yescharacteristic
74C2×#11765848—21no—
75C22×C71⋊C8—21no—
76C2×C8×D71—21no—
77#14556122—21no—
78#14556120C2×C411yescharacteristic
79#14556116C2×C411yescharacteristic
80#14556111—21no—
81#14556107C2×C411yescharacteristic
82#14556102C2×C411yescharacteristic
83C4×Dic142C2311yescharacteristic
84C4×D284C2311yescharacteristic
85C4×C22⋊D71—21no—
86#14556085—21no—
87#14556061C2×C411yescharacteristic
88#14556061C2×C411yescharacteristic
89#14556059C2×C411yescharacteristic
90#14556059C2×C411yescharacteristic
91C4⋊(C71⋊C8)—21no—
92C4⋊(C71⋊C8)C2311yescharacteristic
93C4⋊(C71⋊C8)C2311yescharacteristic
94Dic71⋊C8—21no—
95Dic71⋊C8—21no—
96#14556052C2×C411yescharacteristic
97#14556052C2×C411yescharacteristic
98#14556052C2×C411yescharacteristic
99#14556052C2×C411yescharacteristic
100#14556051C2×C411yescharacteristic
101#14556051C2×C411yescharacteristic
102#14556048C2×C411yescharacteristic
103#14556048C2×C411yescharacteristic
104(C2×D71)⋊C8—21no—
105(C2×D71)⋊C8—21no—
106#14556039C2×C411yescharacteristic
107#14556039C2×C411yescharacteristic
108#14556038—21no—
109C4×C71⋊C8—21no—
110C4×C71⋊C8C2311yescharacteristic
111C4×C71⋊C8C2311yescharacteristic
112C4×C71⋊C8C2311yescharacteristic
113C4×C71⋊C8C2311yescharacteristic
114C8×Dic71—21no—
115C4⋊Q8×C71C2×C411yescharacteristic
116C4⋊Q8×C71C2×C411yescharacteristic
117#125×C71—21no—
118C22⋊Q8×C71—21no—
119C22⋊Q8×C71C2×C411yescharacteristic
120C22⋊Q8×C71C2×C411yescharacteristic
121C4×Q8×C71—21no—
122C4×Q8×C71C2311yescharacteristic
123C4×Q8×C71C2311yescharacteristic
124#117×C71—21no—
125C2×C42×C71C2311yescharacteristic
126C4⋊C8×C71—21no—
127C4⋊C8×C71—21no—
128C4⋊C8×C71—21no—
129C4⋊C8×C71—21no—
130C4⋊C8×C71D411yescharacteristic
131C4⋊C8×C71D411yescharacteristic
132C4⋊C8×C71C2311yescharacteristic
133C4⋊C8×C71C2311yescharacteristic
134C22⋊C8×C71—21no—
135C22⋊C8×C71D411yescharacteristic
136C22⋊C8×C71D411yescharacteristic
137#97×C71—21no—
138#97×C71—21no—
139#97×C71—21no—
140C4×C8×C71—21no—
141C4×C8×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144C2×Dic142C22×C411yescharacteristic
145C2×D284C22×C411yescharacteristic
146C2×C22⋊D71—21no—
147C2×C4×D71—21no—
148#11765855—22no—
149#11765855—22no—
150#11765852—22no—
151#11765852—22no—
152#11765849—22no—
153#11765849—22no—
154#11765849—81no—
155#11765848—81no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8—21no—
159C2×C71⋊C8—21no—
160C2×C71⋊C8—42no—
161C2×C71⋊C8C22×C411yescharacteristic
162C2×C71⋊C8C22×C411yescharacteristic
163C2×C71⋊C8C22×C411yescharacteristic
164C2×C71⋊C8C22×C411yescharacteristic
165C8×D71—81no—
166C4⋊Dic71C22×C411yescharacteristic
167C4⋊Dic71C22×C411yescharacteristic
168Dic71⋊C4—21no—
169Dic71⋊C4—21no—
170Dic71⋊C4—21no—
171Dic71⋊C4—21no—
172C22⋊Dic71—21no—
173(C2×D71)⋊C4—21no—
174(C2×D71)⋊C4—21no—
175(C2×D71)⋊C4—21no—
176(C2×D71)⋊C4—21no—
177C4×Dic71—21no—
178C2×Q8×C71—21no—
179C2×Q8×C71C22×C411yescharacteristic
180C2×Q8×C71C22×C411yescharacteristic
181C22×C4×C71C22×C411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71#4111yescharacteristic
184C2×C8×C71—21no—
185C2×C8×C71—21no—
186C2×C8×C71C2×D411yescharacteristic
187C2×C8×C71C2×D411yescharacteristic
188C2×C8×C71C2×D411yescharacteristic
189C2×C8×C71C2×D411yescharacteristic
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71C22×C411yescharacteristic
198C4⋊C4×C71C22×C411yescharacteristic
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71C22×C411yescharacteristic
204C42×C71C2×D411yescharacteristic
205C42×C71#4111yescharacteristic
206C42×C71C2411yescharacteristic
207C23×C71C4×D411yescharacteristic
208C22×D71—21no—
209Q8×C71—21no—
210Q8×C71—21no—
211Q8×C71—21no—
212Q8×C71—21no—
213Q8×C71—81no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—41no—
223C2×C4×C71C4×D411yescharacteristic
224C2×C4×C71C4×D411yescharacteristic
225C2×C4×C71C4×D411yescharacteristic
226C2×C4×C71C23×C411yescharacteristic
227C2×C4×C71C22×D411yescharacteristic
228C2×C4×C71C2×#4111yescharacteristic
229C22⋊D71—81no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233D284—21no—
234D284—21no—
235C4×D71—41no—
236C4×D71—41no—
237Dic142—21no—
238Dic142—21no—
239C8×C71—21no—
240C8×C71—21no—
241C8×C71—21no—
242C8×C71—21no—
243C8×C71—41no—
244C8×C71—41no—
245C71⋊C8—22no—
246C71⋊C8—21no—
247C71⋊C8—21no—
248C71⋊C8—41no—
249C71⋊C8—41no—
250C71⋊C8—41no—
251C71⋊C8#13112yes—
252C71⋊C8#13112yes—
253C22×C71—21no—
254C22×C71—21no—
255C22×C71—21no—
256C22×C71C2×C4×D411yescharacteristic
257C2×D71—41no—
258C2×D71—41no—
259C2×D71—41no—
260C4×C71—21no—
261C4×C71—21no—
262C4×C71—21no—
263C4×C71—21no—
264C4×C71—41no—
265C4×C71—41no—
266C4×C71—41no—
267C4×C71—41no—
268C4×C71C2×#13111yescharacteristic
269C4×C71C2×#13111yescharacteristic
270C4×C71#57711yescharacteristic
271C4×C71#57811yescharacteristic
272Dic71—41no—
273Dic71—41no—
274Dic71—41no—
275#17969—711nomaximal, Sylow
276C2×C71—41no—
277C2×C71#293111yescharacteristic
278C2×C71#294711yescharacteristic
279C2×C71#295611yescharacteristic
280D71—81no—
281#3172—711no—
282#2993—711no—
283#2985D7111yescharacteristic, complemented
284#1593—711no—
285#1592—711no—
286#1591—711no—
287#1590—711no—
288#1579—711no—
289SD16⋊C8—711no—
290C8×Q16—711no—
291C8×SD16—711no—
292#1569—711no—
293#1494—711no—
294#1493—711no—
295C8×M16—711no—
296C71#1796911yescharacteristic, complemented, Sylow
297#533C2×D7111yescharacteristic
298#529—711no—
299#501—711no—
300#497—711no—
301#491—711no—
302#489—711no—
303#470—711no—
304#470—711no—
305#468—711no—
306#468—711no—
307#446—21no—
308#446—21no—
309#446—711no—
310#446C2×D7111yescharacteristic
311#446C2×D7111yescharacteristic
312C8×Q8—711no—
313C8×Q8C2×D7111yescharacteristic
314C8×Q8C2×D7111yescharacteristic
315C4×Q16—711no—
316C4×Q16—711no—
317C4×SD16—711no—
318C4×SD16—711no—
319#436—711no—
320#436—1421no—
321#435—1421no—
322C8×D4—711no—
323#433—21no—
324#433—21no—
325#433—1421no—
326#432C2×D7111yescharacteristic
327#431C2×D7111yescharacteristic
328#424—1421no—
329#423—711no—
330C4×M16—711no—
331C2×C4×C8—711no—
332#335—711no—
333#335—711no—
334#334—711no—
335#334—711no—
336Q8⋊C8—711no—
337Q8⋊C8—711no—
338Q8⋊C8—711no—
339Q8⋊C8—711no—
340Q8⋊C8—711no—
341Q8⋊C8—711no—
342D4⋊C8—711no—
343D4⋊C8—711no—
344#322—711no—
345#322—711no—
346C82—711no—
347C82—711no—
348C2×Q16—711no—
349C2×Q16—711no—
350C2×SD16—711no—
351C2×SD16—711no—
352C2×M16—711no—
353C2×M16—711no—
354C2×M16—1421no—
355C2×M16—1421no—
356C22×C8—1421no—
357C22×C8—1421no—
358C4⋊Q8C4×D7111yescharacteristic
359C4⋊Q8C4×D7111yescharacteristic
360#126—1421no—
361#125—21no—
362#125—711no—
363#124—711no—
364#123—1421no—
365C22⋊Q8—21no—
366C22⋊Q8—711no—
367C22⋊Q8C4×D7111yescharacteristic
368C22⋊Q8C4×D7111yescharacteristic
369#121—711no—
370C4×Q8—21no—
371C4×Q8—711no—
372C4×Q8C22×D7111yescharacteristic
373C4×Q8C22×D7111yescharacteristic
374C4×D4—711no—
375C4×D4—1421no—
376#117—21no—
377#117—1421no—
378C2×C42C22×D7111yescharacteristic
379#107—711no—
380#107—711no—
381#106—711no—
382#106—711no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—711no—
388C4⋊C8—711no—
389C4⋊C8—1421no—
390C4⋊C8—1421no—
391C4⋊C8—1421no—
392C4⋊C8C22⋊D7111yescharacteristic
393C4⋊C8C22⋊D7111yescharacteristic
394C4⋊C8C22×D7111yescharacteristic
395C4⋊C8C22×D7111yescharacteristic
396#103—711no—
397#103—711no—
398#103—711no—
399#103—711no—
400#103—711no—
401#103—711no—
402#102—711no—
403#102—711no—
404C22⋊C8—21no—
405C22⋊C8—1421no—
406C22⋊C8—1421no—
407C22⋊C8C22⋊D7111yescharacteristic
408C22⋊C8C22⋊D7111yescharacteristic
409#97—21no—
410#97—21no—
411#97—21no—
412#97—711no—
413#97—711no—
414#97—1421no—
415C4×C8—21no—
416C4×C8—21no—
417C4×C8—711no—
418C4×C8—711no—
419C4×C8—711no—
420C4×C8—711no—
421C4×C8—1421no—
422C4×C8—1421no—
423C4×C8C22×D7111yescharacteristic
424C4×C8C22×D7111yescharacteristic
425C2×Q8—21no—
426C2×Q8—711no—
427C2×Q8C2×C4×D7111yescharacteristic
428C2×Q8C2×C4×D7111yescharacteristic
429C2×D4—711no—
430C2×D4—1421no—
431C22×C4—1421no—
432C22×C4C2×C4×D7111yescharacteristic
433C22×C4C2×C22⋊D7111yescharacteristic
434C22×C4#1176586511yescharacteristic
435Q16—1422no—
436Q16—1422no—
437SD16—1422no—
438SD16—1422no—
439M16—5681no—
440M16—5681no—
441M16—1422no—
442M16—1422no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—2842no—
446C2×C8—5681no—
447C2×C8—711no—
448C2×C8—711no—
449C2×C8—711no—
450C2×C8—711no—
451C2×C8—1421no—
452C2×C8—1421no—
453C2×C8—1421no—
454C2×C8—1421no—
455C2×C8C2×C22⋊D7111yescharacteristic
456C2×C8C2×C22⋊D7111yescharacteristic
457C2×C8C2×C22⋊D7111yescharacteristic
458C2×C8C2×C22⋊D7111yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—711no—
467C4⋊C4—711no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4C2×C4×D7111yescharacteristic
473C4⋊C4C2×C4×D7111yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1421no—
477C22⋊C4—1421no—
478C22⋊C4—1421no—
479C22⋊C4—1421no—
480C22⋊C4—1421no—
481C42—21no—
482C42—21no—
483C42—1421no—
484C42C2×C4×D7111yescharacteristic
485C42C2×C22⋊D7111yescharacteristic
486C42#1176586511yescharacteristic
487C42C23×D7111yescharacteristic
488C23—1421no—
489C23C4×C22⋊D7111yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—1421no—
496Q8—1421no—
497D4—5681no—
498D4—1421no—
499D4—1421no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—2841no—
510C2×C4—2841no—
511C2×C4—1421no—
512C2×C4—1421no—
513C2×C4—1421no—
514C2×C4C4×C22⋊D7111yescharacteristic
515C2×C4C4×C22⋊D7111yescharacteristic
516C2×C4C4×C22⋊D7111yescharacteristic
517C2×C4C22×C4×D7111yescharacteristic
518C2×C4C22×C22⋊D7111yescharacteristic
519C2×C4C2×#1176586511yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—2841no—
527C8—2841no—
528C8—2841no—
529C8—712no—
530C8—712no—
531C8—1422no—
532C8—1421no—
533C8—1421no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—2841no—
538C22—2841no—
539C22—2841no—
540C22C2×C4×C22⋊D7111yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—2841no—
550C4—2841no—
551C4—2841no—
552C4#131×D7111yescharacteristic
553C4#131×D7111yescharacteristic
554C4#1798680711yescharacteristic
555C4#1798681411yescharacteristic
556C2—41no—
557C2—5681no—
558C2#2259705011yescharacteristic
559C2#2259718311yescharacteristic
560C2#2259724711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#2985C2
socleC22×C71C2×C4×D4
prime radical#14556102C2×C4
cosocleC42×C71C24
hypercenter#2985D71
hypocenterC71#17969
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2985

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22587247C2semidirect
4. #22587240C2semidirect
5. #22587235C2semidirect
6. #22587230C2semidirect
7. #22587157C2semidirect
8. #22587150C2semidirect
9. C8×#11765855C2semidirect
10. C8×#11765852C2semidirect
11. #22587108C2semidirect
13. #22586795C2semidirect
14. #22586621C2semidirect
15. C71×#2985C2semidirect
283. #2985D71semidirect
296. C71#17969semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225989402
#225974912
#225872472
#225872402
#225872352
#225872302
#225871572
#225871502
C8×#117658552
C8×#117658522
#225871082
#225867982
#225867952
#225866212
C71×#29852
#1796971